Causal structure
In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold.
In modern physics (especially general relativity) spacetime is represented by a Lorentzian manifold.
The causal relations between points in the manifold are interpreted as describing which events in spacetime can influence which other events.
The causal structure of an arbitrary (possibly curved) Lorentzian manifold is made more complicated by the presence of curvature.
Discussions of the causal structure for such manifolds must be phrased in terms of smooth curves joining pairs of points.
Conditions on the tangent vectors of the curves then define the causal relationships.
) then the nonzero tangent vectors at each point in the manifold can be classified into three disjoint types.
We say that a tangent vector is non-spacelike if it is null or timelike.
The canonical Lorentzian manifold is Minkowski spacetime, where
The names for the tangent vectors come from the physics of this model.
The causal relationships between points in Minkowski spacetime take a particularly simple form because the tangent space is also
and hence the tangent vectors may be identified with points in the space.
is the constant representing the universal speed limit, and
The classification of any vector in the space will be the same in all frames of reference that are related by a Lorentz transformation (but not by a general Poincaré transformation because the origin may then be displaced) because of the invariance of the metric.
To do this we first define an equivalence relation on pairs of timelike tangent vectors.
There are then two equivalence classes which between them contain all timelike tangent vectors at the point.
Physically this designation of the two classes of future- and past-directed timelike vectors corresponds to a choice of an arrow of time at the point.
The future- and past-directed designations can be extended to null vectors at a point by continuity.
A Lorentzian manifold is time-orientable[1] if a continuous designation of future-directed and past-directed for non-spacelike vectors can be made over the entire manifold.
is a nondegenerate interval (i.e., a connected set containing more than one point) in
is the image of a path or, more properly, an equivalence class of path-images related by re-parametrisation, i.e. homeomorphisms or diffeomorphisms of
is time-orientable, the curve is oriented if the parameter change is required to be monotonic.
ensure that closed causal curves (such as those consisting of a single point) are not automatically admitted by all spacetimes.
If the manifold is time-orientable then the non-spacelike curves can further be classified depending on their orientation with respect to time.
is These definitions only apply to causal (chronological or null) curves because only timelike or null tangent vectors can be assigned an orientation with respect to time.
Looking at the definitions of which tangent vectors are timelike, null and spacelike we see they remain unchanged if we use
It follows from this that the causal structure of a Lorentzian manifold is unaffected by a conformal transformation.
In various spaces: If a geodesic terminates after a finite affine parameter, and it is not possible to extend the manifold to extend the geodesic, then we have a singularity.
The absolute event horizon is the past null cone of the future timelike infinity.
It is generated by null geodesics which obey the Raychaudhuri optical equation.
